Decidability and Coincidence of Equivalences for
Concurrency

Sibylle Fröschle

Abstract:

There are two fundamental problems concerning
equivalence relations in concurrency. One is: for which system
classes is a given equivalence decidable? The second is: when do
two equivalences coincide? Two well-known equivalences are history
preserving bisimilarity (hpb) and hereditary history preserving
bisimilarity (hhpb). These are both `independence' equivalences:
they reflect causal dependencies between events. Hhpb is obtained
from hpb by adding a `back-tracking' requirement. This seemingly
small change makes hhpb computationally far harder: hpb is
well-known to be decidable for finite-state systems, whereas the
decidability of hhpb has been a renowned open problem for several
years; only recently it has been shown undecidable. The main aim of
this thesis is to gain insights into the decidability problem for
hhpb, and to analyse when it coincides with hpb; less technically,
we might say, to analyse the power of the interplay between
concurrency, causality, and conflict.

We first examine the backtracking condition, and
see that it has two dimensions: the number of transitions over
which one may backtrack, and the number of backtracking moves.
These dimensions translate into two hierarchies of bisimilarities;
we find that both of them are strict, and that each of their levels
is decidable.

Our second approach is to analyse which
behavioural properties of concurrent systems are crucial to the
increased power of hhpb. After establishing a minimum of
behavioural situations necessary to keep hpb and hhpb distinct, we
study two aspects of the interplay of causality, concurrency, and
conflict: three synchronization witness (SW) situations, and the
notion of confusion. With the help of a composition and
decomposition result we prove that in their entirety the SW
situations are essential for non-coincidence (for bounded-degree
systems). However, we show this is not so for confusion, which
disproves the long-standing conjecture that hpb and hhpb coincide
for confusion-free systems.

We continue by studying two structural system
classes with promising behavioural properties. First we consider
basic parallel processes (BPP), with a suitable partial order
semantics. These systems are infinite-state, but they restrict
synchronization. Using the tableau technique, we prove the
decidability and coincidence of hpb and hhpb for simple BPP (SBPP).
The two bisimilarities do not coincide for the complete BPP class,
but we separately achieve decidability of both (a known result for
hpb, but not for hhpb).

The second structural class is (safe) free
choice systems, an important class in Petri net theory. These
systems have a controlled interplay of concurrency and conflict,
and thereby exclude confusion. Having shown that hpb and hhpb do
not coincide here, we identify another interesting candidate: live
strictly state machine decomposable (SSMD) free choice systems. For
this class, we prove that an auxiliary bisimilarity satisfies a
restricted backtracking property. As a consequence we achieve the
coincidence of hpb and hhpb for a subclass of live SSMD free choice
systems: the only known positive result for a class with a
reasonable amount of interplay between concurrency, causality, and
conflict while still admitting considerable nondeterminism.